Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters

(1)

Report

Reference

Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters

SCAILLET, Olivier

Abstract

We study a test statistic based on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 3. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.

SCAILLET, Olivier. Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters. 2005

Available at:

http://archive-ouverte.unige.ch/unige:5759

Disclaimer: layout of this document may differ from the published version.

(2)

2005.05

FACULTE DES SCIENCES

ECONOMIQUES ET SOCIALES

HAUTES ETUDES COMMERCIALES

Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters

O. SCAILLET

(3)

Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters

Olivier Scaillet

HEC Genève and Swiss Finance Institute Université de Genève

Bd Carl Vogt, 102 CH - 1211 Genève 4, Suisse

[email protected]

First version: May 2005

(4)

Abstract

We study a test statistic based on the integrated squared diﬀerence between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 1. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.

Key words and phrases: Nonparametric, Copula density, Goodness-of-fit test, U- statistic.

JEL Classification: C12, D81, G10, G21, G22.

MSC 2000: 62G10, 62P05.

(5)

1 Introduction

Goodness-of-fit (gof) tests have a long history in statistics (D’Agostino and Stephens [4]) while copula models find new interesting applications in finance and insurance (see, e.g., McNeil, Frey and Embrechts [19], Denuit, Dhaene, Goovaerts and Kaas [6]) beside their long acclaimed applications in reliability and survival analysis. Nevertheless relatively little is known about properties of gof tests for copulas despite an obvious need for such tools in applied work; see however Genest, Quessy and Rémillard [15], and Chen, Fan and Patton [3]

for study of gof tests based on the integral probability transformation. The main reason is the technical difficulty induced by the probabilistic behaviour of the empirical copula process (see, e.g., van der Vaart and Wellner [23], Fermanian, Radulovic and Wegkamp [11]). In order to circumvent this difficulty Fermanian [10] suggests to use a gof test based on the integrated weighted squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. In particular he shows that the test statistic is asymptoticallly normally distributed when the bandwidth shrinks to zero. Fan [7] has previously established similar results for bias-corrected gof tests of standard pdf via kernel methods.

In this paper we study the asymptotic properties of the gof test introduced by Fermanian [10], but holding the smoothing parameters fixed and the weight function equal to one as in Fan [9]. We start with recalling the form of the test statistic in Section 2. We derive its interpretation in terms of a weighted integrated squared diﬀerence between the characteristic function of the empirical copula process and the characteristic function of the estimated parametric copula of the null hypothesis. A direct consequence of such an interpretation is test consistency for fixed smoothing parameters. We show that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 1. This is to be contrasted with the result for standard pdf obtained by Fan [9], namely asymptotic properties driven by aU-statistic of order 2 with degeneracy of order 1. To work with copulas carries here a price in terms of analytical tractability of the asymptotic distribution. Therefore for practical implementation we recommend to compute the critical values through a semiparametric bootstrap. Monte Carlo results of Section 3 reveal that the bootstrap performs well in small samples. In particular size and power are less sensitive to smoothing parameter choice than the same characteristics under the asymptotic approximation obtained for a vanishing bandwidth.

2 Test statistic and asymptotic properties

We consider a setting made of i.i.d. observations {Xi = (Xi1, . . . , Xid)⁰;i= 1, . . . , n} of a random vector taking values inR^d. The distribution of X is denoted byF, and the margins are denoted by Fj, j = 1, ..., d. The copula function is denoted byC, and its density byc.

Following Deheuvels [5], let us define the empirical copula function by C(u) :=ˆ 1

n Xn

i=1

I{Fˆ1(Xi1)≤u1, . . . ,Fˆd(Xid)≤ud}, u= (u1, . . . , ud)⁰ ∈[0,1]^d,

whereFˆj is the empirical cumulative distribution functions computed from{Xij;i= 1, . . . , n},

(6)

j = 1, ..., d. Observe that Cˆ is actually a function of the ranks of the observations since nFˆj(Xij)is the rank of Xij among X1j, . . . , Xnj.

Nonparametric estimation procedures for the density of a copula function have been pro- posed by Behnen, Huskova, and Neuhaus [2], and Gijbels and Mielniczuk [17]. They rely on a kernel smoothing directly applied to the transformed sample nYˆi = ( ˆF1(Xi1), . . . ,Fˆd(Xid))⁰;i= 1, . . . , no

. The kernel estimator of the copula density at pointu is simply

ˆ

c(u) := 1 n

Xn i=1

KH(u−Yˆi), (2.1)

where KH(y) = K(H⁻¹y)/detH, K is a d-dimensional kernel, and H is a nonsingular, symmetric matrix of smoothing parameters.

Fermanian [10] suggests to use the following gof test statistic for the parametric family C(u;θ) with density c(u;θ)and θ∈Θ⊂R^p:

J(w) :=ˆ Z h ˆ

c(u)−KH ∗c(u; ˆθ)i2

w(u)du, where ∗ denotes convolution, andw is a weight function.

The estimator θˆcan be computed by a semiparametric maximum likelihood method as in Genest, Ghoudi and Rivest [14], and Shih and Louis [22]. Let θ0 be the true value of the parameter under the null hypothesis of well specification. Then the semiparametric estimator θˆsatisfies under the null hypothesis:

θˆ−θ0 =n⁻¹A(θ0) Xn

i=1

B(Xi;θ0) +op(n⁻^1/2),

whereA(θ0)is ap×ppositive definite matrix andB(Xi;θ0)is ap×1random vector, such that E[B(Xi;θ0)] = 0 andE[B(Xi;θ0)⁰B(Xi;θ0)]<∞.

The following lemma justifies the use ofJˆeven if the bandwidth is not assumed to shrink to zero when the sample size grows to infinity. The kernelK is chosen so that it is symmetric about zero, square integrable, and admits a Fourier transform which vanishes on a set of Lebesgue measure zero.

Lemma 2.1. For w= 1,

Jˆ:= ˆJ(1) = Z

|C(t)¯ˆ −C(t; ˆ¯ θ)|²K¯²(Ht)dt,

where C(t) :=¯ˆ R

exp(it⁰u) ˆC(du), C(t; ˆ¯ θ) := R

exp(it⁰u)C(du; ˆθ), and K(t) := (2π)¯ ⁻^d/2R

exp(it⁰u)K(u)du.

The lemma states thatJˆreduces to the comparison of two empirical characteristic functions when the weight function is equal to one (see Fan [8] for use of empirical characteristic functions in gof tests). The first term C(t)¯ˆ is the empirical characteristic function of the

2

(7)

transformed sample, while the second term is the estimated characteristic function under the parametric assumption. Decomposing the characteristic function C¯ of the copula C in its real part ReC(t) =¯ R

cos(t⁰u)C(du)and its imaginary part ImC(t) =¯ R

sin(t⁰u)C(du), and recognizing that the maps which assign to each copula function C the real and imaginary parts are linear maps, we get from the delta method (van der Vaart and Wellner [23] Section 3.9) that ReC(t), Im¯ˆ C(t), Re¯ˆ C(t; ˆ¯ θ), and ImC(t; ˆ¯ θ), are consistent and jointly asymptotically normally distributed. Hence we deduce that C(t)¯ˆ and C(t; ˆ¯ θ) are consistent and asymptotically normally distributed in the complex plane (see Feueverger and Mureika [12]

for the results on the standard empirical characteristic function). Now recall that a copula function is a cdf on the unit cube, and that there is a one-to-one correspondence between distribution functions and characteristic functions. This gives that no matter the choice of the bandwidth matrix H, the limit J of Jˆwhen w= 1 is such thatJ ≥0 andJ = 0 iﬀthe copula function is well specified, provided that the set {t ∈ R^d : ¯K(t) = 0} has Lebesgue measure zero. Hence we may conclude as in Fan [9] that a test based onJˆis consistent when holding the smoothing parameters fixed and the weight function equal to one.

The following proposition describes the asymptotic behaviour of Jˆwhen the bandwidth matrix does not vanish.

Proposition 2.2. Under the null hypothesis of well specification the asymptotic behaviour of Jˆis that of a U-statistic of order 4 with degeneracy of order 1.

Contrary to the case of a non-vanishing bandwidth for standard pdf (Fan [9]) the asymptotic behaviour of the gof test statistic is not that of a U-statistic of order 2 but of order 4. From the degeneracy of order 1 the rate of convergence is still n, and the exact form of the asymptotic distribution is an infinite sum of weighted chi-square random variables. The weights can be computed in theory from the eigenvalues of an integral equation (see Lee [18]

p. 80). However the kernel of the U-statistic of order 4 involves 24 terms (see the proof of the proposition), and the dimension of the integral is 3. This casts doubt on the numerical accuracy of such a method in practice.

Hereafter we rely on a semiparametric bootstrap to compute the critical values of the test.

First we draw from the estimated copulaC(u; ˆθ)in order to impose the dependence structure of the null hypothesis, and then we use the ranks of these draws to build the bootstrap transformed sample. This semiparametric bootstrap is already exploited in Genest, Quessy and Rémillard [15] since the distributions of their test statistics depend on the unknown parameter value, even in the limit. Its validity for a broad class of gof testing problems is shown in Genest and Rémillard [16]. In particular in the context of gof tests for copulas they show that the sequence associated with the empirical copula is regular for the parametric copula family of the null hypothesis (Genest and Rémillard [16] Proposition 4.2). They also show the regularity of the parametric estimators we use in this paper (Genest and Rémillard [16] Example 4.4). Since we work with linear maps the consistency of the semiparametric bootstrap in our setting is a straightforward consequence of their results.

(8)

3 Monte Carlo results

In this section we study the performance of kernel based gof tests for copulas in small samples when the weight function is kept equal to one. We compare the performance of rejection rules based on 1) asymptotic sets of the chi-square test statistic derived in Fermanian [10]

Corollary 4 for a vanishing bandwidth, 2) sets computed with a bootstrap procedure for the same test statistic, 3) sets computed with a bootstrap procedure for the test staticticnJ.ˆ

First we examine gof tests for the Frank copula. Table I gathers results concerning the size of the diﬀerent testing procedures, i.e., when the true copula is a Frank copula.

This parametric family is often used in actuarial and financial applications, and is easy to draw from; see, e.g., Genest [13] or Nelsen [20]. The chosen values of the parameter are θ ∈ {1,2,3}. They match low to moderate positive dependences as exhibited by the corresponding true values of the Kendall tau, τ ∈{.11, .21, .31}.

The sample sizes are fixed atn= 50 andn= 200. The first sample size can be thought as rather small since we face a bivariate inference problem. For the sake of interpretation samples are generated with both margins corresponding to an exponential distribution with a unit parameter. This can be seen as mimicking the behaviour of claim or duration data.

Note that the numerical results below remain exactly the same if we use uniform margins or other margins with strictly monotonic continuously diﬀerentiable cdf (such as Gaussian or Student margins to mimickfinancial returns) and keep the same seeds in the pseudo-random generators. The reason is that the testing procedures rely intrinsically on ranks.

The kernel estimator of the copula density is based on a bivariate quartic product kernel.

Then the Scott’s rule of thumb (Scott [21]) to select the smoothing parameters gives H = 2.6073n⁻^1/6Σˆ^1/2, where Σˆ is the estimated covariance matrix of the transformed data. To gauge the impact of the choice of the smoothing parameters we report sizes for multiples of this bandwith matrix, namely δH with δ ∈ {.1, .25, .5,1,1.5}. In our simulations the diagonal terms of the selected H are close to one third. We use a bivariate Gauss-Legendre quadrature with 12×12 knots to compute the test statistic, and the optmum routine of the Gauss statistical software with user-supplied analytical gradient and Hessian to optimize the semiparametric loglikelihood. Since we use a Gauss-Legendre quadrature with knots belonging to (0,1)² the restriction that the support of w should be strictly inside the unit square (Fermanian [10] Assumption T) for the asymptotic distribution to be theoretically valid has no practical impact here. The number of bootstrap samples to approximate the p-value is set equal to 500. For each case 200 Monte Carlo simulations are performed, and the rejection rates are computed for each method w.r.t. the conventional significance level of α= 0.05. Programs are available on request.

The results in Table I show that the asymptotic testing procedure is highly sensitive to the bandwidth choice, and that the size distortion may be large. Both bootstrap methods do not suﬀer from these inconvenient features when n= 200. However they tend to slightly underreject when n = 50. From bootstrap theory on higher-order improvements, we know that the bootstrap is expected to yield better results when applied to asymptotic pivots.

However the diﬀerence between both simulation based methods is not striking in our finite sample experiments. We might thus prefer to use the second bootstrap method, namely the one relying simply onn²Jˆ, in light of its computational ease and speed. This second method does not require computing complicated asymptotic bias and variance terms.

4

(9)

In Table II, we gather some results about the power of the testing procedures. We consider a case where 50% of the observed sample is substituted for data drawn from a Student copula with 4 degrees of freedom and a .95 correlation parameter. Again the results indicate that the asymptotic testing procedure is much more aﬀected by the bandwidth choice than both bootstrap procedures. Note further that the smoothing parameters should not be chosen too large (oversmoothing) or too small (undersmoothing) in order to improve on power. We may then conclude that, since the power may be weak in some cases, it is even more crucial to get a well controlled size via the bootstrap.

To get further insight on the behavior of the testing procedure we have also considered a non-Archimedean copula. Tables III and IV correspond to Tables I and II but with the Gaussian copula replacing the Frank copula. The semiparametric and nonparametric estimation methods are kept the same. The chosen parameter values of the Gaussian copula areθ ∈{.17, .32, .47} so that τ ∈{.11, .21, .31}. Results are akin to those got for the Frank copula, and conclusions remain unchanged.

Finally Table V allows for comparison with the Cramer-von Mises (CVM) and Kolmogorov- Smirnov (KS) testing procedures of Genest, Quessy and Rémillard (GQR) [15]. Their procedures are very easy to implement for parametric copula models admitting a distribution function of the probability integral transformation in closed form. This is the case for Archimedean copulas, such as the Frank copula. We can see that results on size are similar to those reported in Table I, but results on power seem to be better in Table II when δ∈{.25, .5,1}. Of course these results are not extensive enough to have a definitive answer (if possible) about which gof test for copulas to favour overall. Even if such an extensive Monte Carlo study is certainly of interest, this is beyond the scope of this note.

(10)

TABLE I: Impact of bandwidth choice on size Rejection rates at 5% level with 200 replications

Pseudo copula: Frank, True copula: Frank

Size n= 50 n= 200

F: θ = 1 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .44 .00 .00 .00 .00 .51 .06 .00 .00 .00 As. Boot. .05 .03 .04 .03 .00 .06 .05 .06 .05 .05 Boot. .05 .03 .05 .04 .04 .06 .05 .07 .05 .06 F: θ = 2 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .50 .00 .00 .00 .00 .51 .06 .01 .00 .00 As. Boot. .05 .06 .05 .02 .00 .06 .06 .06 .06 .04 Boot. .05 .07 .05 .03 .03 .06 .06 .06 .06 .06 F: θ = 3 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .51 .01 .00 .00 .00 .48 .04 .01 .00 .00 As. Boot. .05 .06 .05 .03 .03 .03 .03 .06 .05 .05 Boot. .05 .06 .06 .04 .03 .03 .03 .06 .05 .05

TABLE II: Impact of bandwidth choice on power Rejection rates at 5% level with 200 replications

Pseudo copula: Frank, True copula: mixture of Frank and Student

Power n= 50 n= 200

F: θ = 1 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .51 .08 .02 .00 .00 .65 .77 .77 .04 .00 As. Boot. .06 .20 .26 .09 .00 .13 .71 .96 .72 .18 Boot. .06 .19 .26 .18 .17 .14 .71 .96 .69 .26 F: θ = 2 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .46 .04 .00 .00 .00 .58 .67 .52 .01 .00 As. Boot. .07 .11 .22 .05 .00 .12 .58 .86 .50 .11 Boot. .06 .12 .21 .12 .13 .12 .57 .83 .47 .16 F: θ = 3 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .49 .00 .00 .00 .00 .64 .54 .26 .00 .00 As. Boot. .04 .10 .15 .04 .00 .09 .39 .61 .25 .07 Boot. .04 .10 .13 .10 .08 .10 .39 .60 .21 .12

6

(11)

TABLE III: Impact of bandwidth choice on size Rejection rates at 5% level with 200 replications Pseudo copula: Gaussian, True copula: Gaussian

Size n= 50 n= 200

F: θ =.17 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .46 .00 .00 .00 .00 .48 .04 .00 .00 .00 As. Boot. .03 .02 .06 .04 .00 .03 .08 .06 .06 .04 Boot. .06 .04 .06 .05 .06 .05 .08 .07 .06 .05 F: θ =.32 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .51 .00 .00 .00 .00 .62 .03 .01 .00 .00 As. Boot. .04 .02 .04 .02 .02 .07 .05 .05 .05 .04 Boot. .04 .04 .04 .04 .06 .09 .07 .08 .05 .05 F: θ =.47 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .69 .00 .00 .00 .00 .73 .04 .01 .00 .00 As. Boot. .02 .03 .06 .01 .01 .04 .04 .06 .05 .03 Boot. .06 .05 .06 .06 .06 .06 .05 .07 .06 .06

TABLE IV: Impact of bandwidth choice on power Rejection rates at 5% level with 200 replications

Pseudo copula: Gaussian, True copula: mixture of Gaussian and Student

Power n= 50 n= 200

F: θ =.17 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .72 .01 .02 .01 .00 .64 .49 .85 .35 .00 As. Boot. .06 .01 .38 .07 .01 .02 .39 1 1 .72

Boot. .07 .22 .39 .23 .12 .16 .92 1 1 .88

F: θ =.32 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .76 .02 .01 .00 .00 .75 .25 .52 .13 .00 As. Boot. .06 .01 .29 .03 .00 .01 .08 .96 .96 .45 Boot. .07 .20 .34 .18 .10 .12 .80 .99 .97 .69 F: θ =.47 δ =.1 .25 .5 1 1.5 δ=.1 .25 .5 1 1.5 Asym. .81 .09 .00 .00 .00 .83 .05 .21 .03 .00 As. Boot. .03 .03 .17 .02 .00 .03 .01 .72 .77 .21 Boot. .09 .12 .21 .14 .09 .10 .65 .96 .85 .48

(12)

TABLE V: Comparison with CVM and KS tests of GQR Rejection rates at 5% level with 200 replications

Size/Power: Pseudo copula: Frank, True copula: Frank/mixture

Size n= 50 n= 200 Power n= 50 n= 200

CVM KS CVM KS CVM KS CVM KS

F:θ = 1 .05 .03 .06 .05 F: θ = 1 .06 .04 .28 .23 F:θ = 2 .04 .03 .06 .05 F: θ = 2 .04 .04 .24 .19 F:θ = 3 .04 .03 .06 .05 F: θ = 3 .05 .04 .22 .16

8

(13)

APPENDIX

A Proof of Lemma 2.1

The Fourier transform ofc(u)ˆ −KH ∗c(u; ˆθ)is given by (2π)⁻^d/2R

exp(it⁰u)h ˆ

c(u)−KH ∗c(u; ˆθ)i du

= (2π)⁻^d/2R

exp(it⁰u)hR

KH(u−v)n

C(dv)ˆ −C(dv; ˆθ)oi du

=R

exp(it⁰v) ¯K(Ht)n

C(dv)ˆ −C(dv; ˆθ)o .

The stated result is then deduced from an application of Parseval’s identity toJˆand the last equality.

B Proof of Proposition 2.2

Let us introduce {Yi = (F1(Xi1), . . . , Fd(Xid))⁰;i= 1, . . . , n}. ObviouslyC is the cdf ofY. We consider

Iˆ= Z "

1 n

Xn i=1

a(u,Xi)

#2

du, (B.1)

with

a(u,Xi) = KH(u−Yi) +K_H⁽¹⁾(u−Yi)⁰( ˆYi−Yi)

− Z

KH(u−v)C(dv;θ0)−μ(u)A(θ0)B(Xi;θ0), where K_H⁽¹⁾ denotes the first derivative of KH, and μ(u) =R

KH(u−v)∂C(dv;θ0)/∂θ⁰. As in Fan [9] Lemma 3.1, via simple second-order Taylor expansions of ˆc(u) aroundYi, and c(u; ˆθ) around θ0, we may check that n( ˆJ −I) =ˆ op(1) under the null hypothesis and our assumptions since √

n(ˆθ−θ0) =Op(1)and√

n( ˆYi −Yi) = Op(1). Indeed substituting the expansions into Jˆand collecting terms yieldn( ˆJ−I) =ˆ nP7

i=1Iî with Iî = R Iî(u)du and

Iˆ1(u) = 1 n²

Xn i=1

Xn j=1

h

KH(u−Yi) +K_H⁽¹⁾(u−Yi)⁰( ˆYi−Yi)i h

( ˆYj −Yj)⁰K_H⁽²⁾(u−Yj)( ˆYj−Yj)i ,

Iˆ2(u) = 1 4n²

Xn i=1

Xn j=1

h

( ˆYi −Yi)⁰K_H⁽²⁾(u−Yi)( ˆYi−Yi)i h

( ˆYj −Yj)⁰K_H⁽²⁾(u−Yj)( ˆYj−Yj) i

,

(14)

Iˆ3(u) = Z Z h

KH(u−v)³

C(dv;θ0) +∂C(dv;θ0)/∂θ⁰(ˆθ−θ0)´i h

KH(u−s)(ˆθ−θ0)⁰∂²C(ds;θ0)/∂θ∂θ⁰(ˆθ−θ0)i ,

Iˆ4(u) = 1 4

Z Z h

KH(u−v)(ˆθ−θ0)⁰∂²C(dv;θ0)/∂θ∂θ⁰(ˆθ−θ0)i h

KH(u−s)(ˆθ−θ0)⁰∂²C(ds;θ0)/∂θ∂θ⁰(ˆθ−θ0)i ,

Iˆ5(u) = −1 n

Xn i=1

h

KH(u−Yi) +K_H⁽¹⁾(u−Yi)⁰( ˆYi−Yi)i

∙Z

KH(u−v)(ˆθ−θ0)⁰∂²C(dv;θ0)/∂θ∂θ⁰(ˆθ−θ0)

¸ ,

Iˆ6(u) = −1 n

Xn i=1

h

( ˆYi−Yi)⁰K_H⁽²⁾(u−Yi)( ˆYi−Yi) i

∙Z

KH(u−v)³

C(dv;θ0) +∂C(dv;θ0)/∂θ⁰(ˆθ−θ0)´¸

,

Iˆ7(u) = − 1 2n

Xn i=1

h

( ˆYi −Yi)⁰K_H⁽²⁾(u−Yi)( ˆYi −Yi)i

∙Z

KH(u−v)(ˆθ−θ0)⁰∂²C(dv;θ0)/∂θ∂θ⁰(ˆθ−θ0)

¸ .

Then since 1 n

Xn i=1

KH(u−Yi)− Z

KH(u−v)C(dv;θ0)converges to zero and the absolute value of each of its elements is bounded for any u, we can deduce that n( Î1 + Î6) = op(1) andn( Î3+ Î5) =op(1), while nIˆ2 =Op(n⁻¹), nIˆ4 =Op(n⁻¹), and nIˆ7 =Op(n⁻¹).

Now we can rewrite Equation (B.1) as Iˆ= 1

n⁴ Xn

i=1

Xn j=1

Xn k=1

Xn l=1

g(Xi,Xj,Xk,Xl), (B.2) with

g(Xi,Xj,Xk,Xl) = Z

{KH(u−Yi) +K_H⁽¹⁾(u−Yi)⁰(I[Xk ≤Xi]−Yi)

− Z

KH(u−v)C(dv;θ0)−μ(u)A(θ0)B(Xi;θ0)} {KH(u−Yj) +K_H⁽¹⁾(u−Yj)⁰(I[Xl ≤Xj]−Yj)

− Z

KH(u−v)C(dv;θ0)−μ(u)A(θ0)B(Xj;θ0)}du,

10

(15)

with I[Xk≤Xi] = (I[Xk1 ≤Xi1], ..., I[Xkd≤Xid])⁰.

By symmetrization of the kernel g (Lee [18] p. 7) we know that Iˆ shares the same asymptotic behaviour as the V-statistic:

Vˆ = 1 n⁴

Xn i=1

Xn j=1

Xn k=1

Xn l=1

ψ(Xi,Xj,Xk,Xl),

where ψ is a symmetric kernel given by:

24ψ(Xi,Xj,Xk,Xl) = g(Xi,Xj,Xk,Xl) +g(Xi,Xj,Xl,Xk) +g(Xi,Xk,Xj,Xl) +g(Xi,Xk,Xl,Xj) +g(Xi,Xl,Xj,Xk) +g(Xi,Xl,Xk,Xj) +g(Xj,Xi,Xk,Xl) +g(Xj,Xi,Xl,Xk) +g(Xj,Xk,Xi,Xl) +g(Xj,Xk,Xl,Xi) +g(Xj,Xl,Xi,Xk) +g(Xj,Xl,Xk,Xi) +g(Xk,Xi,Xj,Xl) +g(Xk,Xi,Xl,Xj) +g(Xk,Xj,Xi,Xl) +g(Xk,Xj,Xl,Xi) +g(Xk,Xl,Xi,Xj) +g(Xk,Xl,Xj,Xi) +g(Xl,Xi,Xj,Xk) +g(Xl,Xi,Xk,Xj) +g(Xl,Xj,Xi,Xk) +g(Xl,Xj,Xk,Xi) +g(Xl,Xk,Xi,Xj) +g(Xl,Xk,Xj,Xi).

Through the connection between aV-statisticVˆ of order 4 and its associatedU-statistics Uˆj of order j = 1, ...,4 (Lee [18] p. 183) we can write

Vˆ =S4⁽⁴⁾

(n−1)(n−2)(n−3)

n³ Uˆ4+S4⁽³⁾

(n−1)(n−2)

n³ Uˆ3+S4⁽²⁾

(n−1)

n³ Uˆ2+S4⁽¹⁾

1 n³Uˆ1, where the Stirling numbers of the second kind are given by S4⁽⁴⁾ = 1, S4⁽³⁾ = 6, S4⁽²⁾ = 7, S4⁽¹⁾ = 1 (Abramowitz and Stegun [1] p. 835). Hence the asymptotic behaviour of Iˆis that of the leading U-statistic Uˆ4 of order 4, whose kernel is equal toψ(Xi,Xj,Xk,Xl).

To determine the degree of degeneracy when the null hypothesis holds true, we need to investigate the nullity of appropriate conditional expectations ofψ, namely

σ₁² = Var[ψ1(Xi)], σ₂² = Var[ψ2(Xi,Xj)], σ₃² = Var[ψ3(Xi,Xj,Xk)], where

ψ1(xi) = E[ψ(Xi,Xj,Xk,Xl)|Xi =xi],

ψ2(xi,xj) = E[ψ(Xi,Xj,Xk,Xl)|Xi =xi,Xj =xj],

ψ3(xi,xj,xk) = E[ψ(Xi,Xj,Xk,Xl)|Xi =xi,Xj =xj,Xk=xk].

We can see that ψ1(xi) = 0 under the null hypothesis, so that σ²₁ = 0. On the contrary ψ2(xi, ,xj) 6= 0 since E[g(Xi,Xj,Xk,Xl)|Xi = xi,Xj = xj] 6= 0, for example, so that σ₂² >0. Hence the degree of degeneracy is 1, andnUˆ4 has a nondegenerate limit distribution (Lee [18] p. 90). This yields the stated result.

(16)

Acknowledgements

Olivier Scaillet thanks the editor and the three reviewers for constructive criticism and care- ful reading which lead to improving and correcting earlier versions of the paper. He also thanks Marcelo Fernandes and Patrick Gagliardini for helpful discussions. He has received support by the Swiss National Science Foundation through the National Center of Compe- tence: Financial Valuation and Risk Management. Part of this research was done when he was visiting THEMA and ECARES.

Downloadable at

http://www.hec.unige.ch/professeurs/SCAILLET_Olivier/pages_web/Home_Page_of_Olivier_Scaillet.htm

12

(17)

References

[1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.

[2] K. Behnen, M. Huskova, G. Neuhaus, Rank estimators of scores for testing indepen- dence, Statistics and Decision 3 (1985) 239-262.

[3] X. Chen, Y. Fan, A. Patton, Simple tests for models of dependence between multiple financial time series, with applications to US equity returns and exchange rates, Working paper LSE (2003).

[4] R. D’Agostino, M. Stephens, Goodness-of-Fit Techniques, Marcel Dekker, New York, 1986.

[5] P. Deheuvels, La fonction de dépendance empirique et ses propriétés: Un test non para- métrique d’indépendance, Bulletin de l’Académie royale de Belgique: Classe des sciences 65 (1979) 274-292.

[6] M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas, Actuarial Theory for Dependent Risks, Wiley, New York, 2005.

[7] Y. Fan, Testing the goodness-of-fit of a parametric density function by kernel methods, Econometric Theory 10 (1994) 316-356.

[8] Y. Fan, Goodness-of-fit tests for a multivariate distribution by the empirical characteristic function, Journal of Multivariate Analysis 62 (1997) 36-63.

[9] Y. Fan, Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters, Econometric Theory 14 (1998) 604-621.

[10] J.-D. Fermanian, Goodness of fit tests for copulas, Journal of Multivariate Analysis 95 (2005) 119-152.

[11] J.-D. Fermanian, D. Radulovic, M. Wegkamp, Weak convergence of empirical copula processes, Bernoulli 10 (2004) 847-860.

[12] A. Feueverger, R. Mureika, The empirical characteristic function and its applications, Annals of Statistics 5 (1977) 88-97.

[13] C. Genest, Frank’s family of bivariate distributions, Biometrika 74 (1987) 549-555.

[14] C. Genest, K. Ghoudi, L.-P. Rivest, A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika 82 (1998) 543-552.

[15] C. Genest, J.-F. Quessy, B. Rémillard, Goodness-of-fit procedures for copula models based on the probability integral transformation, Scandinavian Journal of Statistics forthcoming (2006).

(18)

[16] C. Genest, B. Rémillard, Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models, Working paper Université Laval (2005).

[17] I. Gijbels, J. Mielniczuk, Estimating the density of a copula function, Communications in Statistics: Theory and Methods 19 (1990) 445-464.

[18] A. Lee, U-Statistics: Theory and Practice, Marcel Dekker, New York, 1990.

[19] A. McNeil, R. Frey, P. Embrechts, Quantitative Risk Management: Concepts, Tech- niques and Tools, Princeton University Press, Princeton, 2005.

[20] R. Nelsen, An Introduction to Copulas, Lecture Notes in Statistics, Springer-Verlag, New York, 1999.

[21] D. Scott, Multivariate Density Estimation. Theory, Practice and Visualization, Wiley, New York, 1992.

[22] J. Shih, T. Louis, Inferences on the association parameter in copula models for bivariate survival data, Biometrics 51 (1995) 1384-1399.

[23] A. van der Vaart, J. Wellner, Weak Convergence and Empirical Processes, Springer- Verlag, New York, 1996.

14